Tim: I don't think that no spark on one side would cause the engine to die and not restart. The engine employs a "wasted spark" ignition system, so it should run about as well on one cylinder as two (I know this to be a fact), it just will lack horsepower. So, if one side only is dropping out, then it would cause a precipitous, but not deadly, drop in power output, that is, if mowing, all of a sudden it would struggle, but otherwise be OK.
As far as heating up and konking out, and it being an ignition problem, you should check the air gap to be sure it is correct (a matchbook cover or 0.010" of an inch). I would also check to be sure you have the correct heat range in the plugs, Kohler calls for Champion RV17YC or equivalent in the M18/M20 motors gapped at 0.035" The plugs do make a difference, Matt Gonitzke found a corresponding NKG plug, and he promised the number to anyone who asked. I found an AC Delco plug in mine and the "equivalent" I was able to source improved performance a lot, as I recall.
The engine should still run if the plugs are functional and not fouled; the correct heat range would hedge against their fouling.
All that being said, the ignition module, i.e., Magneto, is a common point of failure with these engines, so I doubt the money would be wasted should you decide to replace it.
Kirk Engines (above) sports an aftermarket part which outperforms the original, as I understand it.
Good luck.